# Hybrid iterative scheme for a generalized equilibrium problems, variational inequality problems and fixed point problem of a finite family of κ i -strictly pseudocontractive mappings

## Abstract

In this article, by using the S-mapping and hybrid method we prove a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of κ i -strictly pseudocontractive mappings and the set of generalized equilibrium defined by Ceng et al., which is a solution of two sets of variational inequality problems. Moreover, by using our main result we have a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of κ i -strictly pseudocontractive mappings and the set of solution of a finite family of generalized equilibrium defined by Ceng et al., which is a solution of a finite family of variational inequality problems.

## 1 Introduction

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. A mapping T of H into itself is called nonexpansive if Tx - Tyx - y for all x, y H. We denote by F(T) the set of fixed points of T (i.e., F(T) = {x H : Tx = x}) Goebel and Kirk [1] showed that F(T) is always closed convex, and also nonempty provided T has a bounded trajectory.

Recall the mapping T is said to be κ-strict pseudo-contration if there exist κ [0, 1) such that

${∥Tx-Ty∥}^{2}\le {∥x-y∥}^{2}+\kappa {∥\left(I-T\right)x-\left(I-T\right)y∥}^{2}\forall x,y\in D\left(T\right).$
(1.1)

Note that the class of κ-strict pseudo-contractions strictly includes the class of nonexpansive mappings, that is T is nonexpansive if and only if T is 0-strict pseudo-contractive. If κ = 1, T is said to be pseudo-contraction mapping. T is strong pseudo-contraction if there exists a positive constant λ (0, 1) such that T + λI is pseudo-contraction. In a real Hilbert space H (1.1) is equivalent to

$⟨Tx-Ty,x-y⟩\le {∥x-y∥}^{2}-\frac{1-\kappa }{2}{∥\left(I-T\right)x-\left(I-T\right)y∥}^{2}\phantom{\rule{2.77695pt}{0ex}}\forall x,y\in D\left(T\right).$
(1.2)

T is pseudo-contraction if and only if

$⟨Tx-Ty,x-y⟩\le {∥x-y∥}^{2}\phantom{\rule{1em}{0ex}}\forall x,y\in D\left(T\right).$

T is strong pseudo-contraction if there exists a positive constant λ (0, 1)

$⟨Tx-Ty,x-y⟩\le \left(1-\lambda \right){∥x-y∥}^{2}\phantom{\rule{1em}{0ex}}\forall x,y\in D\left(T\right)$

The class of κ-strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contraction mappings and class of strong pseudo-contraction mappings is independent of the class of κ-strict pseudo-contraction.

A mapping A of C into H is called inverse-strongly monotone, see [2] if there exists a positive real number α such that

$⟨x-y,Ax-Ay⟩\ge \alpha {∥Ax-Ay∥}^{2}$

for all x, y C.

The equilibrium problem for G is to determine its equilibrium points, i.e., the set

$EP\left(G\right)=\left\{x\in G:G\left(x,y\right)\ge 0,\phantom{\rule{1em}{0ex}}\forall y\in C\right\}.$
(1.3)

Given a mapping T : CH, let G(x, y) = 〈Tx, y - x〉 for all x, y C. Then, z EP(F) if and only if 〈Tz, y - z〉 ≥ 0 for all y C, i.e., z is a solution of the variational inequality. Let A : CH be a nonlinear mapping. The variational inequality problem is to find a u C such that

$⟨v-u,Au⟩\ge 0$
(1.4)

for all v C. The set of solutions of the variational inequality is denoted by VI(C, A).

In 2005, Combettes and Hirstoaga [3] introduced some iterative schemes of finding the best approximation to the initial data when EP(G) is nonempty and proved strong convergence theorem.

Also in [3] Combettes and Hiratoaga, following [4] define S r : HC by

${S}_{r}\left(x\right)=\left\{z\in C:G\left(z,y\right)+\frac{1}{r}⟨y-z,z-x⟩\ge 0\forall y\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\in C\right\}.$
(1.5)

hey proved that under suitable hypotheses G, S r is single-valued and firmly nonexpansive with F(S r ) = EP(G).

Numerous problems in physics, optimization, and economics reduce to find a element of EP(G) (see, e.g., [516])

Let CB(H) be the family of all nonempty closed bounded subsets of H and $ℋ\left(.,.\right)$ be the Hausdorff metric on CB(H) defined as

$ℋ\left(U,V\right)=\text{max}\left\{\underset{u\in U}{\text{sup}}d\left(u,V\right),\underset{v\in V}{\text{sup}}d\left(U,v\right)\right\},\phantom{\rule{1em}{0ex}}\forall U,V\in CB\left(H\right),$

where d(u, V) = infvVd(u, v), d(U, v) = infuUd(u, v), and d(u, v) = u - v.

Let C be a nonempty closed convex subset of H. Let φ : C be a real-valued function, T : CCB(H) a multivalued mapping and Φ : H × C × C an equilibrium-like function, that is, Φ(w, u, v) + Φ(w, v, u) = 0 for all (w, u, v) H × C × C which satisfies the following conditions with respect to the multivalued map T : CCB(H).

(H 1) For each fixed v C, (ω, u) Φ(ω, u, v) is an upper semicontinuous function from H × C to , that is, for (ω, u) H × C, whenever ω n ω and u n u as n → ∞,

$\underset{n\to \infty }{\text{lim sup}}\Phi \left({\omega }_{n},{u}_{n},v\right)\le \Phi \left(\omega ,u,v\right);$

(H 2) For each fixed (w, v) H × C, u Φ(w, u, v) is a concave function;

(H 3) For each fixed (w, u) H × C, v Φ(w, u, v) is a convex function.

In 2009, Ceng et al. [17] introduced the following generalized equilibrium problem (GEP) as follows:

$\left(\text{GEP}\right)\left\{\begin{array}{c}\text{Find}\phantom{\rule{2.77695pt}{0ex}}u\in C\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}w\in T\left(u\right)\phantom{\rule{2.77695pt}{0ex}}\text{such}\phantom{\rule{2.77695pt}{0ex}}\text{that}\hfill \\ \Phi \left(w,u,v\right)+\phi \left(v\right)-\phi \left(u\right)\ge 0,\phantom{\rule{2.77695pt}{0ex}}\forall v\in C.\hfill \end{array}\right\$
(1.6)

The set of such solutions u C of (GEP) is denote by (GEP) s (Φ, φ).

In the case of φ ≡ 0 and Φ(w, u, v) ≡ G(u, v), then (GEP) s (Φ, φ) is denoted by EP(G). By using Nadler's theorem they introduced the following algorithm:

Let x1 C and w1 T(x1), there exists sequences {w n } H and {x n }, {u n } C such that

$\left\{\begin{array}{c}{w}_{n}\in T\left({x}_{n}\right),∥{w}_{n}-{w}_{n+1}∥\le \left(1+\frac{1}{n}\right)ℋ\left(T\left({x}_{n}\right),T\left({x}_{n+1}\right)\right),\hfill \\ \Phi \left({w}_{n},{u}_{n},v\right)+\phi \left(v\right)-\phi \left({u}_{n}\right)+\frac{1}{{r}_{n}}⟨{u}_{n}-{x}_{n},v-{u}_{n}⟩\ge 0,\phantom{\rule{2.77695pt}{0ex}}\forall u\in C,\hfill \\ {x}_{n+1}={\alpha }_{n}f\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)S{u}_{n},\phantom{\rule{1em}{0ex}}n=1,2,....\hfill \end{array}\right\$
(1.7)

They proved a strong convergence theorem of the sequence {x n } generated by (1.7) as follows:

Theorem 1.1. (See [17] ) Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H and let φ : C be a lower semicontinuous and convex functional. Let T : CCB(H) be $ℋ$-Lipschitz continuous with constant μ, Φ : H × C × C be an equilibrium-like function satisfying (H1)-(H3) and S be a nonexpansive mapping of C into itself such that $F\left(S\right)\cap {\left(GEP\right)}_{s}\left(\Phi ,\phi \right)\ne \varnothing$. Let f be a contraction of C into itself and let {x n }, {w n }, and {u n } be sequences generated by (1.7), where {α n } [0,1] and {r n } (0, ∞) satisfy

If there exists a constant λ > 0 such that

$\Phi \left({w}_{1},{T}_{{r}_{1}}\left({x}_{1}\right),{T}_{{r}_{2}}\left({x}_{2}\right)\right)+\Phi \left({w}_{2},{T}_{{r}_{2}}\left({x}_{2}\right),{T}_{{r}_{1}}\left({x}_{1}\right)\right)\le -\lambda {∥{T}_{{r}_{1}}\left({x}_{1}\right)-{T}_{{r}_{2}}\left({x}_{2}\right)∥}^{2}$
(1.8)

for all (r1, r2) Ξ × Ξ,(x1, x2) C × C and w i T(x i ), i = 1, 2, where Ξ = {r n : n ≥ 1}, then for $\stackrel{^}{x}={P}_{F\left(S\right)\cap {\left(GEP\right)}_{s}\left(\Phi ,\phi \right)}f\left(\stackrel{^}{x}\right)$, there exists $\stackrel{^}{w}\in T\left(\stackrel{^}{x}\right)$ such that $\left(\stackrel{^}{x},\stackrel{^}{w}\right)$ is a solution of (GEP) and

${x}_{n}\to \stackrel{^}{x},\phantom{\rule{2.77695pt}{0ex}}{w}_{n}\to \stackrel{^}{w}\phantom{\rule{2.77695pt}{0ex}}and\phantom{\rule{2.77695pt}{0ex}}{u}_{n}\to \stackrel{^}{x}\phantom{\rule{2.77695pt}{0ex}}as\phantom{\rule{2.77695pt}{0ex}}n\to \infty .$

In 2011, Kangtunyakarn [18] proved the following theorem for strict pseudocontractive mapping in Hilbert space by using hybrid method as follows:

Theorem 1.2. Let C be a nonempty closed convex subset of a Hilbert space H. Let F and G be bifunctions from C × C into satisfying (A1)-(A4), respectively. Let A : CH be a α-inverse strongly monotone mapping and let B : CH be a β-inverse strongly monotone mapping. Let T : CC be a κ-strict pseudo-contraction mapping with $F=F\left(T\right)\cap EP\left(F,A\right)\cap EP\left(G,B\right)\ne \varnothing$. Let {x n } be a sequence generated by x1 C = C1 and

$\left\{\begin{array}{c}F\left({u}_{n},u\right)+\left(A{x}_{n},u-{u}_{n}\right)+\frac{1}{{r}_{n}}⟨u-{u}_{n},{u}_{n}-{x}_{n}⟩\ge 0,\phantom{\rule{1em}{0ex}}\forall u\in C,\hfill \\ G\left({v}_{n},v\right)+\left(B{x}_{n},v-{v}_{n}\right)+\frac{1}{{s}_{n}}⟨v-{v}_{n},{v}_{n}-{x}_{n}⟩\ge 0,\phantom{\rule{1em}{0ex}}\forall v\in C,\hfill \\ {z}_{n}={\delta }_{n}{u}_{n}+\left(1-{\delta }_{n}\right){v}_{n}\hfill \\ {y}_{n}={\alpha }_{n}{z}_{n}+\left(1-{\alpha }_{n}\right)T{z}_{n}\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:∥{y}_{n}-z∥\le ∥{x}_{n}-z∥\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}\forall n\ge 1,\hfill \end{array}\right\$
(1.9)

where ${\left\{{\alpha }_{n}\right\}}_{n=0}^{\infty }$ is sequence in [0,1], r n [a, b] (0, 2α) and s n [c, d] (0, 2β) satisfy the following condition:

$\begin{array}{c}\left(i\right)\underset{n\to \infty }{\text{lim}}{\delta }_{n}=\delta \in \left(0,1\right)\\ \left(ii\right)\phantom{\rule{2.77695pt}{0ex}}0\le \kappa \le {\alpha }_{n}<1,\phantom{\rule{1em}{0ex}}\forall n\ge 1\end{array}$

Then x n converges strongly to ${P}_{F}{x}_{1}$.

From motivation of (1.7) and (1.9), we define the following algorithm as follows:

Algorithm 1.3. Let T i , i = 1,2,...,N, be κ i -pseudo contraction mappings of C into itself and κ = max{κ i : i = 1,2,..., N} and let S n be the S-mappings generated by T1, T2, ..., T N and ${\alpha }_{1}^{\left(n\right)},{\alpha }_{2}^{\left(n\right)},...,{\alpha }_{N}^{\left(n\right)}$ where ${\alpha }_{j}^{\left(n\right)}=\left({\alpha }_{1}^{n,j},{\alpha }_{2}^{n,j},{\alpha }_{3}^{n,j}\right)\in I×I×I,I=\left[0,1\right],{\alpha }_{1}^{n,j}+{\alpha }_{2}^{n,j}+{\alpha }_{3}^{n,j}=1$ and $\kappa for all $j=1,2,...,N-1,\kappa for all j = 1,2,...,N. Let x1 C = C1 and ${w}_{1}^{1}\in T\left({x}_{1}\right),{w}_{1}^{2}\in D\left({x}_{1}\right)$, there exists sequence $\left\{{w}_{n}^{1}\right\},\left\{{w}_{n}^{2}\right\}\in H$ and {x n }, {u n }, {v n } C such that

$\left\{\begin{array}{c}{w}_{n}^{1}\in T\left({x}_{n}\right),\phantom{\rule{1em}{0ex}}∥{w}_{n}^{1}-{w}_{n+1}^{1}∥\le \left(1+\frac{1}{n}\right)ℋ\left(T\left({x}_{n}\right),T\left({x}_{n+1}\right)\right),\hfill \\ {w}_{n}^{2}\in D\left({x}_{n}\right),\phantom{\rule{2.77695pt}{0ex}}∥{w}_{n}^{2}-{w}_{n+1}^{2}∥\le \left(1+\frac{1}{n}\right)ℋ\left(D\left({x}_{n}\right),D\left({x}_{n+1}\right)\right),\hfill \\ {\Phi }_{1}\left({w}_{n}^{1},{u}_{n},u\right)+{\phi }_{1}\left(u\right)-{\phi }_{1}\left({u}_{n}\right)+\frac{1}{{r}_{n}}⟨{u}_{n}-{x}_{n},u-{u}_{n}⟩\ge 0,\phantom{\rule{1em}{0ex}}\forall u\in C,\hfill \\ {\Phi }_{2}\left({w}_{n}^{2},{v}_{n},v\right)+{\phi }_{2}\left(v\right)-{\phi }_{2}\left({v}_{n}\right)+\frac{1}{{s}_{n}}⟨{v}_{n}-{x}_{n},v-{v}_{n}⟩\ge 0,\phantom{\rule{1em}{0ex}}\forall v\in C,\hfill \\ {z}_{n}={\delta }_{n}{P}_{C}\left(I-\lambda A\right){u}_{n}+\left(1-{\delta }_{n}\right){P}_{C}\left(I-\eta B\right){v}_{n},\hfill \\ {y}_{n}={\alpha }_{n}{z}_{n}+\left(1-{\alpha }_{n}\right){S}_{n}{z}_{n},\hfill \\ {C}_{n+1}=\left\{z\in {C}_{n}:∥{y}_{n}-z∥\le ∥{x}_{n}-z∥\right\},\hfill \\ {x}_{n+1}={P}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}\forall n\ge 1.\hfill \end{array}\right\$
(1.10)

where D, T : CCB(H) are $ℋ$-Lipschitz continuous with constant μ1, μ2, respectively, Φ1, Φ2 : H × C × C are equilibrium-like functions satisfying (H 1)-(H 3), A : CH is a α-inverse strongly monotone mapping and B : CH is a β-inverse strongly monotone mapping.

In this article, we prove under some control conditions on {δ n }, {α n }, {s n }, and {r n } that the sequence {x n } generated by (1.7) converges strongly to ${P}_{F}{x}_{1}$ where $F={\cap }_{i=1}^{N}F\left({T}_{i}\right)\cap {\left(GEP\right)}_{s}\left({\Phi }_{1},{\phi }_{1}\right)\cap {\left(GEP\right)}_{s}\left({\Phi }_{2},{\phi }_{2}\right)\cap F\left({G}_{1}\right)\cap F\left({G}_{2}\right)$, G1, G2 : CC are defined by G1(x) = P C (x - λAx), G2(x) = P C (x - ηBx), x C and ${P}_{F}{x}_{1}$ is solution of the following system of variational inequality:

$\left\{\begin{array}{c}⟨A{x}^{*},\phantom{\rule{2.77695pt}{0ex}}x-{x}^{*}⟩\ge 0,\hfill \\ ⟨B{x}^{*},\phantom{\rule{2.77695pt}{0ex}}x-{x}^{*}⟩\ge 0.\hfill \end{array}\right\$

## 2 Preliminaries

In this section, we need the following lemmas and definition to prove our main result.

Let C be a nonempty closed convex subset of H. Then for any x H, there exists a unique nearest point in C, denoted by P C x, such that

$∥x-{P}_{C}x∥\le ∥x-y∥,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}y\in C.$

The following lemma is a property of P C .

Lemma 2.1. (See [19].) Given x H and y C. Then P C x = y if and only if there holds the inequality

$⟨x-y,y-z⟩\ge 0\phantom{\rule{1em}{0ex}}\forall z\in C.$

Lemma 2.2. (See [20] ) Let C be a closed convex subset of a strictly convex Banach space E. Let {T n : n } be a sequence of nonexpansive mappings on C. Suppose ${\cap }_{n=1}^{\infty }F\left({T}_{n}\right)$ is nonempty. Let {λ n } be a sequence of positive numbers with ${\sum }_{n=1}^{\infty }{\lambda }_{n}=1$. Then a mapping S on C defined by

$S\left(x\right)={\sum }_{n=1}^{\infty }{\lambda }_{n}{T}_{n}x$

for x C is well defined, nonexpansive and $F\left(S\right)={\cap }_{n=1}^{\infty }F\left({T}_{n}\right)$ hold.

The following lemma is well known.

Lemma 2.3. Let H be Hilbert space, C be a nonempty closed convex subset of H. Let T : CC be a κ-strictly pseudo-contractive, then the fixed point set F(T) of T is closed and convex so that the projection PF(T)is well defined.

In 2009, Kangtunyakarn and Suantai [21] introduced the S-mapping generated by a finite family of κ-strictly pseudo contractive mappings and real numbers as follows:

Definition 2.1. Let C be a nonempty convex subset of real Hilbert space. Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of κ i -strict pseudo-contractions of C into itself. For each j = 1,2,..., N, let ${\alpha }_{j}=\left({a}_{1}^{j},{\alpha }_{2}^{j},{\alpha }_{3}^{j}\right)\in I×I×I$ where I [0,1] and ${\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1$. We define the mapping S : CC as follows:

$\begin{array}{ll}\hfill {U}_{0}& =I\phantom{\rule{2em}{0ex}}\\ \hfill {U}_{1}& ={\alpha }_{1}^{1}{T}_{1}{U}_{0}+{\alpha }_{2}^{1}{U}_{0}+{\alpha }_{3}^{1}I\phantom{\rule{2em}{0ex}}\\ \hfill {U}_{2}& ={\alpha }_{1}^{2}{T}_{2}{U}_{1}+{\alpha }_{2}^{2}{U}_{1}+{\alpha }_{3}^{2}I\phantom{\rule{2em}{0ex}}\\ \hfill {U}_{3}& ={\alpha }_{1}^{3}{T}_{3}{U}_{2}+{\alpha }_{2}^{3}{U}_{2}+{\alpha }_{3}^{3}I\phantom{\rule{2em}{0ex}}\\ \cdot \phantom{\rule{2em}{0ex}}\\ \cdot \phantom{\rule{2em}{0ex}}\\ \cdot \phantom{\rule{2em}{0ex}}\\ \hfill {U}_{N-1}& ={\alpha }_{1}^{N-1}{T}_{N-1}{U}_{N-2}+{\alpha }_{2}^{N-1}{U}_{N-2}+{\alpha }_{3}^{N-1}I\phantom{\rule{2em}{0ex}}\\ \hfill S& ={U}_{N}={\alpha }_{1}^{N}{T}_{N}{U}_{N-1}+{\alpha }_{2}^{N}{U}_{N-1}+{\alpha }_{3}^{N}I.\phantom{\rule{2em}{0ex}}\end{array}$
(2.1)

This mapping is called S-mapping generated by T1, ..., T N and α1, α2, ..., α N .

Lemma 2.4. (See [21] ) Let C be a nonempty closed convex subset of real Hilbert space. Let ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ be a finite family of κ-strict pseudo contraction mapping of C into C with ${\cap }_{i=1}^{N}F\left({T}_{i}\right)\ne 0̸$ and κ = max{κ i : i = 1, 2,..., N} and let ${\alpha }_{j}=\left({a}_{1}^{j},{\alpha }_{2}^{j},{\alpha }_{3}^{j}\right)\in I×I×I$, j = 1,2,3,...,N, where $I=\left[0,1\right],\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1,\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{1}^{j},\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{3}^{j}\in \left(\kappa ,1\right)$ for all j = 1,2,...,N - 1 and ${\alpha }_{1}^{N}\in \left(\kappa ,1\right],{\alpha }_{3}^{N}\in \left[\kappa ,1\right)\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{2}^{j}\in \left[\kappa ,1\right)$ for all j = 1,2,..., N. Let S be the mapping generated by T1,....,T N and α1, α2,...,α N . Then $F\left(S\right)={\cap }_{i=1}^{N}F\left({T}_{i}\right)$ and S is a nonexpansive mapping.

Lemma 2.5. (See [22] ) Let C be a nonempty closed convex subset of a real Hilbert space H and S : CC be a self-mapping of C. If S is a κ-strict pseudo-contraction mapping, then S satisfies the Lipschitz condition

$∥Sx-Sy∥\le \frac{1+\kappa }{1-\kappa }∥x-y∥,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}y\in C.$

We prove the following lemma by using the concept of the S-mapping as follows:

Lemma 2.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T i , i = 1,2,...,N be κ i strictly pseudo-contraction mappings of C into itself and κ = max{κ i : i = 1,2,...,N} and let ${\alpha }_{j}^{\left(n\right)}=\left({\alpha }_{1}^{n,j},{\alpha }_{2}^{n,j},{\alpha }_{3}^{n,j}\right),\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{j}=\left({\alpha }_{1}^{j},{\alpha }_{2}^{j},{\alpha }_{3}^{j}\right)\in I×I×I$, where $I=\left[0,1\right],\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{1}^{n,j}+{\alpha }_{2}^{n,j}+{\alpha }_{3}^{n,j}=1$ and ${\alpha }_{1}^{j}+{\alpha }_{2}^{j}+{\alpha }_{3}^{j}=1$ such that ${\alpha }_{i}^{n,j}\to {\alpha }_{i}^{j}\in \left[0,1\right]$ as n → ∞ for i = 1, 3 and j = 1,2,3,..., N. For every n , let S and S n be the S-mapping generated by T1, T2,..., T N and α1, α2,...,α N and T1, T2,..., T N and ${\alpha }_{1}^{\left(n\right)},{\alpha }_{2}^{\left(n\right)},\dots ,{\alpha }_{N}^{\left(n\right)}$, respectively. Then limn→∞S n x n - Sx n = 0 for every bounded sequence {x n } in C.

Proof. Let {x n } be bounded sequence in C, U k and Un,kbe generated by T1,T2,...,T N and α1,α2,...,α N and T1,T2,...,T N and ${\alpha }_{1}^{\left(n\right)},{\alpha }_{2}^{\left(n\right)},\dots ,{\alpha }_{N}^{\left(n\right)}$, respectively. For each n , we have

$\begin{array}{ll}\hfill ∥{U}_{n,1}{x}_{n}-{U}_{1}{x}_{n}∥& =∥{\alpha }_{1}^{n,1}{T}_{1}{x}_{n}+\left(1-{\alpha }_{1}^{n,1}\right){x}_{n}-{\alpha }_{1}^{1}{T}_{1}{x}_{n}-\left(1-{\alpha }_{1}^{1}\right){x}_{n}∥\phantom{\rule{2em}{0ex}}\\ =∥{\alpha }_{1}^{n,1}{T}_{1}{x}_{n}-{\alpha }_{1}^{n,1}{x}_{n}-{\alpha }_{1}^{1}{T}_{1}{x}_{n}+{\alpha }_{1}^{1}{x}_{n}∥\phantom{\rule{2em}{0ex}}\\ =∥\left({\alpha }_{1}^{n,1}-{\alpha }_{1}^{1}\right){T}_{1}{x}_{n}-\left({\alpha }_{1}^{n,1}-{\alpha }_{1}^{1}\right){x}_{n}∥\phantom{\rule{2em}{0ex}}\\ =\left|{a}_{1}^{n,1}-{\alpha }_{1}^{1}\right|∥{T}_{1}{x}_{n}-{x}_{n}∥\phantom{\rule{2em}{0ex}}\end{array}$
(2.2)

and for k {2, 3,..., N}, by using Lemma 2.5, we obtain

(2.3)

By (2.2) and (2.3), we have

This together with the assumption ${\alpha }_{i}^{n,j}\to {\alpha }_{i}^{j}$ as n → ∞ (i = 1, 3, j = 1,2,..., N), we can conclude that

$\underset{n\to \infty }{\text{lim}}∥{S}_{n}{x}_{n}-S{x}_{n}∥=0.$

Lemma 2.7. (See [23]) Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and S : CC be a nonexpansive mapping. Then I - S is demi-closed at zero.

Lemma 2.8. (See [24]) Let C be a closed convex subset of H. Let {x n } be a sequence in H and u H. Let q = P C u, if {x n } is such the ω(x n ) C and satisfy the condition

$∥{x}_{n}-u∥\le ∥u-q∥,\phantom{\rule{1em}{0ex}}\forall n\in ℕ.$

Then x n q, as n → ∞.

Definition 2.2. A multivalued map T : CCB(H) is say to be $ℋ$-Lipschitz continuous if there exists a constant μ > 0 such that

$ℋ\left(T\left(u\right)-T\left(v\right)\right)\le \mu ∥u-v∥,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall u,v\in C,$

where $ℋ\left(.,.\right)$ is the Hausdorff metric on CB(H).

Lemma 2.9. (Nadler's theorem, see [25]) Let (X, ) be a normed vector space and $ℋ\left(.,.\right)$ is the Hausdorff metric on CB(H). If U, V CB(X), then for any given ϵ > 0 and u U, there exists v V such that

$∥u-v∥\le \left(1+\epsilon \right)ℋ\left(U,V\right).$

Let C be a nonempty closed convex subset of a real Hilbert space H. Let φ: CH be a real-valued function, T : CCB(H) be a multivalued map and Φ : H × C × C be an equilibrium-like function.

To solve the GEP, let us assume that the equilibrium-like function Φ : H × C × C satisfies the following conditions with respect to the multivalued map T: CCB(H).

(H 1) For each fixed v C, (ω, u) Φ(ω, u, v) is an upper semicontinuous function from H × C to , that is, for (ω, u) H × C, whenever ω n ω and u n u as n → ∞,

$\underset{n\to \infty }{\text{lim sup}}\Phi \left({\omega }_{n},{u}_{n},v\right)\le \Phi \left(\omega ,u,v\right);$

(H 2) For each fixed (w, v) H × C, u Φ(w, u, v) is a concave function;

(H 3) For each fixed (w, u) H × C, v Φ(w, u, v) is a convex function.

Theorem 2.10. (See [17]) Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H, and let φ : C be a lower semicontinuous and convex functional. Let T : CCB(H) be $ℋ$-Lipschitz continuous with constant μ, and Φ : H × C × C be an equilibrium-like function satisfying (H 1)-(H 3). Let r > 0 be a constant. For each x C, take w x T(x) arbitrarily and define a mapping T r : CC as follows:

${T}_{r}\left(x\right)=\left\{u\in C:\Phi \left({w}_{x},u,v\right)+\phi \left(v\right)-\phi \left(u\right)+\frac{1}{r}⟨u-x,v-u⟩\ge 0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall v\in C\right\}.$

Then, there hold the following:

(a) T r is single-valued;

(b) T r is firmly nonexpansive (that is, for any u, v C, T r u - T r v2 ≤ 〈T r u-T r v, u-v〉) if

$\Phi \left({w}_{1},{T}_{r}\left({x}_{1}\right),{T}_{r}\left({x}_{2}\right)\right)+\Phi \left({w}_{2},{T}_{r}\left({x}_{2}\right),{T}_{r}\left({x}_{1}\right)\right)\le 0,$

for all (x1, x2) C × C and all w i T(xi), i = 1,2;

(c) F(T r ) = (GEP) s (Φ, φ)

(d) (GEP) s (Φ, φ) is closed and convex.

Lemma 2.11. (See [26]) Let C be a nonempty closed convex subset of a Hilbert space H and let G : CC be defined by

$G\left(x\right)={P}_{C}\left(x-\lambda Ax\right),\phantom{\rule{1em}{0ex}}\forall x\in C,$

with λ > 0. Then x* VI (C, A) if and only if x* F(G).

## 3 Main results

In this section, we prove a strong convergence theorem of the sequence {x n } generated by (1.10) to ${P}_{F}{x}_{1}$.

Theorem 3.1. Let C be a nonempty bounded, closed, and convex subset of Hilbert space H and let φ1, φ2 : be a lower semicontinuous and convex function. Let D, T : CCB(H) be $ℋ$-Lipschitz continuous with constant μ1, μ2, respectively, Φ12: H × C × C be equilibrium-like functions satisfying (H 1) - (H3). Let A: CH be a α-inverse strongly monotone mapping and B : CH be a β-inverse strongly monotone mapping, let T i , i = 1,2,...,N, be κ i -pseudo contraction mappings of C into itself and κ = max{κ i :i = 1,2,..., N} with $F={\cap }_{i=1}^{N}F\left({T}_{i}\right)\cap {\left(GEP\right)}_{s}\left({\Phi }_{1},{\phi }_{1}\right)\cap {\left(GEP\right)}_{s}\left({\Phi }_{2},{\phi }_{2}\right)\cap F\left({G}_{1}\right)\cap F\left({G}_{2}\right)$, where G1, G2 : CC are defined by G1(x) P C (x-λAx), G2(x) = P C (x-ηBx), x C. Let S n be the S-mappings generated by T1,T2,...,T N and ${\alpha }_{1}^{\left(n\right)},{\alpha }_{2}^{\left(n\right)},...,{\alpha }_{N}^{\left(n\right)}$ where ${\alpha }_{j}^{\left(n\right)}=\left({\alpha }_{1}^{n,j},{\alpha }_{2}^{n,j},{\alpha }_{3}^{n,j}\right)\in I×I×I,I=\left[0,1\right],{\alpha }_{1}^{n,j}+{\alpha }_{2}^{n,j}+{\alpha }_{3}^{n,j}=1$ and $\kappa for all $j=1,2,...,N-1,\kappa for all j = 1,2,...,N and let {x n }, {u n }, {v n }, $\left\{{w}_{n}^{1}\right\}$, and $\left\{{w}_{n}^{2}\right\}$ be sequences generated by (1.10), where {α n } is a sequence in [0,1], r n , λ [a, b] (0, ) and s n , η [c, d] (0, 2β), for every n and suppose the following conditions hold:

(i) $\underset{n\to \infty }{\text{lim}}{\delta }_{n}=\delta \in \left(0,1\right)$,

(ii) 0 ≤ κα n < 1, n ≥ 1,

(iii) $\sum _{n=1}^{\infty }\left|{\alpha }_{1}^{n+1,j}-{\alpha }_{1}^{n,j}\right|<\infty ,\sum _{n=1}^{\infty }\left|{\alpha }_{3}^{n+1,j}-{\alpha }_{3}^{n,j}\right|<\infty$, for all j {1,2,3,...,N}.

(iv) There exists λ1, λ2 such that

$\left\{\begin{array}{c}{\Phi }_{1}\left({w}_{1}^{1},{T}_{{r}_{1}}\left({x}_{1}\right),{T}_{{r}_{2}}\left({x}_{2}\right)\right)+{\Phi }_{1}\left({w}_{2}^{1},{T}_{{r}_{2}}\left({x}_{2}\right),{T}_{{r}_{1}}\left({x}_{1}\right)\right)\le -{\lambda }_{1}{∥{T}_{{r}_{1}}\left({x}_{1}\right)-{T}_{{r}_{2}}\left({x}_{2}\right)∥}^{2}and\hfill \\ {\Phi }_{2}\left({w}_{1}^{2},{T}_{{s}_{1}}\left({x}_{1}\right),{T}_{{s}_{2}}\left({x}_{2}\right)\right)+{\Phi }_{2}\left({w}_{2}^{2},{T}_{{s}_{2}}\left({x}_{2}\right),{T}_{{s}_{1}}\left({x}_{1}\right)\right)\le -{\lambda }_{2}{∥{T}_{{s}_{1}}\left({x}_{1}\right)-{T}_{{s}_{2}}\left({x}_{2}\right)∥}^{2}\hfill \end{array}\right\$
(3.1)

for all $\left({r}_{1},{r}_{2}\right)\in \Theta ×\Theta ,\phantom{\rule{2.77695pt}{0ex}}\left({s}_{1},{s}_{2}\right)\in \Xi ×\Xi ,\phantom{\rule{2.77695pt}{0ex}}{w}_{i}^{1}\in T\left({x}_{i}\right)$ and ${w}_{i}^{2}\in D\left({x}_{i}\right)$, for i = 1,2 where Θ = {r n : n ≥ 1} and Ξ = {s n : n ≥ 1}. Then {x n } converges strongly to ${P}_{F}{x}_{1}$ which is a solution of (3.2):

$\left\{\begin{array}{c}⟨A{x}^{*},\phantom{\rule{2.77695pt}{0ex}}x-{x}^{*}⟩\ge 0,\hfill \\ ⟨B{x}^{*},\phantom{\rule{2.77695pt}{0ex}}x-{x}^{*}⟩\ge 0.\hfill \end{array}\right\$
(3.2)

Proof. From (3.1) for every r Θ, we have

${\Phi }_{1}\left({w}_{1}^{1},{T}_{r}\left({x}_{1}\right),{T}_{r}\left({x}_{2}\right)\right)+{\Phi }_{1}\left({w}_{2}^{1},{T}_{r}\left({x}_{2}\right),{T}_{r}\left({x}_{1}\right)\right)\le -{\lambda }_{1}{∥{T}_{r}\left({x}_{1}\right)-{T}_{r}\left({x}_{2}\right)∥}^{2}\le 0,$
(3.3)

for all (x1, x2) C × C and ${w}_{i}^{1}\in T\left({x}_{i}\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}i=1,2$.

Similarly, for every s Ξ, we have

${\Phi }_{2}\left({w}_{1}^{2},{T}_{s}\left({x}_{1}\right),{T}_{s}\left({x}_{2}\right)\right)+{\Phi }_{2}\left({w}_{2}^{2},{T}_{s}\left({x}_{2}\right),{T}_{s}\left({x}_{1}\right)\right)\le -{\lambda }_{2}{∥{T}_{s}\left({x}_{1}\right)-{T}_{s}\left({x}_{2}\right)∥}^{2}\le 0.$
(3.4)

for all (x1, x2) C × C and ${w}_{i}^{2}\in D\left({x}_{i}\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}i=1,2$. From (3.3) and (3.4), we have Theorem 2.10 hold.

It is easy to see that I - λ A and I - η B are nonexpansive mapping. Indeed, since A is a α-inverse strongly monotone mapping with λ (0, 2α), we have

$\begin{array}{ll}\hfill {∥\left(I-\lambda A\right)x-\left(I-\lambda A\right)y∥}^{2}& ={∥x-y-\lambda \left(Ax-Ay\right)∥}^{2}\phantom{\rule{2em}{0ex}}\\ ={∥x-y∥}^{2}-2\lambda ⟨x-y,Ax-Ay⟩+{\lambda }^{2}{∥Ax-Ay∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le {∥x-y∥}^{2}-2\alpha \lambda {∥Ax-Ay∥}^{2}+{\lambda }^{2}{∥Ax-Ay∥}^{2}\phantom{\rule{2em}{0ex}}\\ ={∥x-y∥}^{2}+\lambda \left(\lambda -2\alpha \right){∥Ax-Ay∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le {∥x-y∥}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$

Thus (I - λA) is nonexpansive, so is I - ηB. Since

${\Phi }_{1}\left({w}_{n}^{1},{u}_{n},u\right)+{\phi }_{1}\left(u\right)-{\phi }_{1}\left({u}_{n}\right)+\frac{1}{{r}_{n}}⟨{u}_{n}-{x}_{n},u-{u}_{n}⟩\ge 0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall u\in C,$

and Theorem 2.10, we have ${u}_{n}={T}_{{r}_{n}}{x}_{n}$. Since

${\Phi }_{2}\left({w}_{n}^{2},{v}_{n},v\right)+{\phi }_{2}\left(v\right)-{\phi }_{2}\left({v}_{n}\right)+\frac{1}{{s}_{n}}⟨{v}_{n}-{x}_{n},v-{v}_{n}⟩\ge 0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall v\in C,$

and Theorem 2.10, we have ${v}_{n}={T}_{{s}_{n}}{x}_{n}$. Let $z\in F$, again by Theorem 2.10, we have $z={T}_{{r}_{n}}z={T}_{{s}_{n}}z={P}_{C}\left(I-\lambda A\right)z={P}_{C}\left(I-\eta B\right)z$. From nonexpansiveness of $\left\{{T}_{{r}_{n}}\right\},\phantom{\rule{2.77695pt}{0ex}}\left\{{T}_{{s}_{n}}\right\},\phantom{\rule{2.77695pt}{0ex}}\left\{I-\lambda A\right\}$, and {I - ηB}, we have

$\begin{array}{ll}\hfill ∥{z}_{n}-z∥& =∥{\delta }_{n}\left({P}_{C}\left(I-\lambda A\right){u}_{n}-z\right)+\left(1-{\delta }_{n}\right)\left({P}_{C}\left(I-\eta B\right){v}_{n}-z\right)∥\phantom{\rule{2em}{0ex}}\\ \le {\delta }_{n}∥{P}_{C}\left(I-\lambda A\right){u}_{n}-z∥+\left(1-{\delta }_{n}\right)∥{P}_{C}\left(I-\eta B\right){v}_{n}-z∥\phantom{\rule{2em}{0ex}}\\ \le {\delta }_{n}∥{T}_{{r}_{n}}{x}_{n}-z∥+\left(1-{\delta }_{n}\right)∥{T}_{{s}_{n}}{x}_{n}-z∥\phantom{\rule{2em}{0ex}}\\ \le ∥{x}_{n}-z∥.\phantom{\rule{2em}{0ex}}\end{array}$
(3.5)

By (3.5), we have

$\begin{array}{ll}\hfill ∥{y}_{n}-z∥& =∥{\alpha }_{n}\left({z}_{n}-z\right)+\left(1-{\alpha }_{n}\right)\left({S}_{n}{z}_{n}-z\right)∥\phantom{\rule{2em}{0ex}}\\ \le {\alpha }_{n}∥{z}_{n}-z∥+\left(1-{\alpha }_{n}\right)∥{S}_{n}{z}_{n}-z∥\phantom{\rule{2em}{0ex}}\\ \le ∥{z}_{n}-z∥\le ∥{x}_{n}-z∥.\phantom{\rule{2em}{0ex}}\end{array}$
(3.6)

Next, we show that C n is closed and convex for every n . It is obvious that C n is closed. In fact, we know that, for z C n ,

$∥{y}_{n}-z∥\le ∥{x}_{n}-z∥\phantom{\rule{2.77695pt}{0ex}}\text{is}\phantom{\rule{2.77695pt}{0ex}}\text{equivalent}\phantom{\rule{2.77695pt}{0ex}}\text{to}\phantom{\rule{2.77695pt}{0ex}}{∥{y}_{n}-{x}_{n}∥}^{2}+2⟨{y}_{n}-{x}_{n},{x}_{n}-z⟩\le 0.$

So, we have that z1, z2 C n and t (0,1), it follows that

$\begin{array}{ll}\hfill {∥{y}_{n}-{x}_{n}∥}^{2}& +2⟨{y}_{n}-{x}_{n},{x}_{n}-\left(t{z}_{1}+\left(1-t\right){z}_{2}\right)⟩\phantom{\rule{2em}{0ex}}\\ =t\left(2⟨{y}_{n}-{x}_{n},{x}_{n}-{z}_{1}⟩+{∥{y}_{n}-{x}_{n}∥}^{2}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\left(1-t\right)\left(2⟨{y}_{n}-{x}_{n},{x}_{n}-{z}_{2}⟩+{∥{y}_{n}-{x}_{n}∥}^{2}\right)\phantom{\rule{2em}{0ex}}\\ \le 0,\phantom{\rule{2em}{0ex}}\end{array}$

then, we have C n is convex. By Theorem 2.10 and Lemma 2.3, we conclude that $F$ is closed and convex. This implies that ${P}_{F}$ is well defined. Next, we show that $F\subset {C}_{n}$ for every n . Putting $q\in F$, by (3.6), it is easy to see that q C n , then we have $F\subset {C}_{n}$ for all n . Since ${x}_{n}={P}_{{C}_{n}}{x}_{1}$, for every w C n , we have

$∥{x}_{n}-{x}_{1}∥\le ∥w-{x}_{1}∥,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\forall n\in ℕ.$
(3.7)

In particular, we have

$∥{x}_{n}-{x}_{1}∥\le ∥{P}_{F}{x}_{1}-{x}_{1}∥.$
(3.8)

Since C is bounded, we have {x n } is bounded, so are {u n }, {v n }, {z n }, and {y n }. Since ${x}_{n+1}={P}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}\subset {C}_{n}$ and ${x}_{n}={P}_{{C}_{n}}{x}_{1}$, we have

$\begin{array}{ll}\hfill 0& \le ⟨{x}_{1}-{x}_{n},{x}_{n}-{x}_{n+1}⟩\phantom{\rule{2em}{0ex}}\\ =⟨{x}_{1}-{x}_{n},{x}_{n}-{x}_{1}+{x}_{1}-{x}_{n+1}⟩\phantom{\rule{2em}{0ex}}\\ \le -{∥{x}_{n}-{x}_{1}∥}^{2}+∥{x}_{n}-{x}_{1}∥∥{x}_{1}-{x}_{n+1}∥,\phantom{\rule{2em}{0ex}}\end{array}$

it implies that

$∥{x}_{n}-{x}_{1}∥\le ∥{x}_{n+1}-{x}_{1}∥.$

Hence, we have limn→∞x n - x1 exists. Since

$\begin{array}{ll}\hfill {∥{x}_{n}-{x}_{n+1}∥}^{2}& ={∥{x}_{n}-{x}_{1}+{x}_{1}-{x}_{n+1}∥}^{2}\phantom{\rule{2em}{0ex}}\\ ={∥{x}_{n}-{x}_{1}∥}^{2}+2⟨{x}_{n}-{x}_{1},{x}_{1}-{x}_{n+1}⟩+{∥{x}_{1}-{x}_{n+1}∥}^{2}\phantom{\rule{2em}{0ex}}\\ ={∥{x}_{n}-{x}_{1}∥}^{2}+2⟨{x}_{n}-{x}_{1},{x}_{1}-{x}_{n}+{x}_{n}-{x}_{n+1}⟩+{∥{x}_{1}-{x}_{n+1}∥}^{2}\phantom{\rule{2em}{0ex}}\\ ={∥{x}_{n}-{x}_{1}∥}^{2}-2{∥{x}_{n}-{x}_{1}∥}^{2}+2⟨{x}_{n}-{x}_{1},{x}_{n}-{x}_{n+1}⟩+{∥{x}_{1}-{x}_{n+1}∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le {∥{x}_{1}-{x}_{n+1}∥}^{2}-{∥{x}_{n}-{x}_{1}∥}^{2},\phantom{\rule{2em}{0ex}}\end{array}$
(3.9)

it implies that

$\underset{n\to \infty }{\text{lim}}∥{x}_{n}-{x}_{n+1}∥=0.$
(3.10)

Since ${x}_{n+1}={P}_{{C}_{n+1}}{x}_{1}\in {C}_{n+1}$, we have

$∥{y}_{n}-{x}_{n+1}∥\le ∥{x}_{n}-{x}_{n+1}∥,$

by (3.10), we have

$\underset{n\to \infty }{\text{lim}}∥{y}_{n}-{x}_{n+1}∥=0.$
(3.11)

Since

$∥{y}_{n}-{x}_{n}∥\le ∥{y}_{n}-{x}_{n+1}∥+∥{x}_{n+1}-{x}_{n}∥,$

by (3.10) and (3.11), we have

$\underset{n\to \infty }{\text{lim}}∥{y}_{n}-{x}_{n}∥=0.$
(3.12)

Next, we show that

$\underset{n\to \infty }{\text{lim}}∥{z}_{n}-{S}_{n}{z}_{n}∥=0.$
(3.13)

By definition of y n , we have

${y}_{n}-{z}_{n}=\left(1-{\alpha }_{n}\right)\left({S}_{n}{z}_{n}-{z}_{n}\right).$
(3.14)

Claim that

$\underset{n\to \infty }{\text{lim}}∥{z}_{n}-{x}_{n}∥=0.$
(3.15)

Putting M n = P C (I - λA)u n and N n = P C (I - ηB)v n , we have

$∥{z}_{n}-{x}_{n}∥\le {\delta }_{n}∥{M}_{n}-{x}_{n}∥+\left(1-{\delta }_{n}\right)∥{N}_{n}-{x}_{n}∥.$
(3.16)

Let $z\in F$. Since ${T}_{{r}_{n}}$ is firmly nonexpansive mapping and ${T}_{{r}_{n}}{x}_{n}={u}_{n}$, we have

$\begin{array}{ll}\hfill {∥z-{u}_{n}∥}^{2}& ={∥{T}_{{r}_{n}}z-{T}_{{r}_{n}}{x}_{n}∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le ⟨{T}_{{r}_{n}}z-{T}_{{r}_{n}}{x}_{n},z-{x}_{n}⟩\phantom{\rule{2em}{0ex}}\\ =\frac{1}{2}\left({∥{u}_{n}-z∥}^{2}+{∥{x}_{n}-z∥}^{2}-{∥{u}_{n}-{x}_{n}∥}^{2}\right).\phantom{\rule{2em}{0ex}}\end{array}$

Hence

${∥{u}_{n}-z∥}^{2}\le {∥{x}_{n}-z∥}^{2}-{∥{u}_{n}-{x}_{n}∥}^{2}.$
(3.17)

Since ${T}_{{r}_{n}}$ is firmly nonexpansive mapping and ${T}_{{s}_{n}}{x}_{n}={v}_{n}$, by using the same method as (3.17), we have

${∥{v}_{n}-z∥}^{2}\le {∥{x}_{n}-z∥}^{2}-{∥{v}_{n}-{x}_{n}∥}^{2}.$
(3.18)

By nonexpansiveness of S n and (3.17), (3.18), we have

$\begin{array}{ll}\hfill {∥{y}_{n}-z∥}^{2}& \le {∥{z}_{n}-z∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le {\delta }_{n}{∥{u}_{n}-z∥}^{2}+\left(1-{\delta }_{n}\right){∥{v}_{n}-z∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le {\delta }_{n}\left({∥{x}_{n}-z∥}^{2}-{∥{u}_{n}-{x}_{n}∥}^{2}\right)+\left(1-{\delta }_{n}\right)\left({∥{x}_{n}-z∥}^{2}-{∥{v}_{n}-{x}_{n}∥}^{2}\right)\phantom{\rule{2em}{0ex}}\\ ={∥{x}_{n}-z∥}^{2}-{\delta }_{n}{∥{u}_{n}-{x}_{n}∥}^{2}-\left(1-{\delta }_{n}\right){∥{v}_{n}-{x}_{n}∥}^{2},\phantom{\rule{2em}{0ex}}\end{array}$

it implies that

$\begin{array}{ll}\hfill {\delta }_{n}{∥{u}_{n}-{x}_{n}∥}^{2}& \le {∥{x}_{n}-z∥}^{2}-{∥{y}_{n}-z∥}^{2}-\left(1-{\delta }_{n}\right){∥{v}_{n}-{x}_{n}∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le {∥{x}_{n}-z∥}^{2}-{∥{y}_{n}-z∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le \left(∥{x}_{n}-z∥+∥{y}_{n}-z∥\right)∥{x}_{n}-{y}_{n}∥,\phantom{\rule{2em}{0ex}}\end{array}$

by (3.12) and condition (i), we have

$\underset{n\to \infty }{\text{lim}}∥{u}_{n}-{x}_{n}∥=0.$
(3.19)

By using the same method as (3.19), we have

$\underset{n\to \infty }{\text{lim}}∥{v}_{n}-{x}_{n}∥=0.$
(3.20)

Since

(3.21)

Claim that

$\underset{n\to \infty }{\text{lim}}∥A{u}_{n}-Az∥=\underset{n\to \infty }{\text{lim}}∥B{v}_{n}-Bz∥=0.$

By nonexpansiveness of P C , we have

$\begin{array}{c}{‖{y}_{n}-z‖}^{2}\le {‖{z}_{n}-z‖}^{2}\\ \le {\delta }_{n}{‖{P}_{C}\left(I-\lambda A\right){u}_{n}-{P}_{C}\left(I-\lambda A\right)z‖}^{2}\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}+\left(1-{\delta }_{n}\right){‖{P}_{C}\left(I-\eta B\right){v}_{n}-{P}_{C}\left(I-\eta B\right)z‖}^{2}\\ \le {\delta }_{n}{‖\left(I-\lambda A\right){u}_{n}-\left(I-\lambda A\right)z‖}^{2}+\left(1-{\delta }_{n}\right){‖\left(I-\eta B\right){v}_{n}-\left(I-\eta B\right)z‖}^{2}\\ \le {\delta }_{n}{‖\left({u}_{n}-\lambda A{u}_{n}-\left(z-\lambda Az\right)‖}^{2}+\left(1-{\delta }_{n}\right){‖\left({v}_{n}-\eta B{v}_{n}-\left(z-\eta Bz\right)‖}^{2}\\ ={\delta }_{n}{‖\left({u}_{n}-z\right)-\lambda \left(A{u}_{n}-Az\right)‖}^{2}+\left(1-{\delta }_{n}\right){‖\left({v}_{n}-z\right)-\eta \left(B{v}_{n}-Bz\right)‖}^{2}\\ ={\delta }_{n}\left({‖{u}_{n}-z‖}^{2}+{\lambda }^{2}{‖A{u}_{n}-Az‖}^{2}-2\lambda 〈{u}_{n}-z,A{u}_{n}-Az〉\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}+\left(1-{\delta }_{n}\right)\left({‖{v}_{n}-z‖}^{2}+{\eta }^{2}{‖B{v}_{n}-Bz‖}^{2}-2\eta 〈{v}_{n}-z,B{v}_{n}-Bz〉\right)\\ \le {\delta }_{n}\left({‖{u}_{n}-z‖}^{2}+{\lambda }^{2}{‖A{u}_{n}-Az‖}^{2}-2\lambda \alpha {‖A{u}_{n}-Az‖}^{2}\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}+\left(1-{\delta }_{n}\right)\left({‖{v}_{n}-z‖}^{2}+{\eta }^{2}{‖B{v}_{n}-Bz‖}^{2}-2\eta \beta {‖B{v}_{n}-Bz‖}^{2}\right)\\ \le {\delta }_{n}\left({‖{x}_{n}-z‖}^{2}+\lambda \left(\lambda -2\alpha \right){‖A{u}_{n}-Az‖}^{2}\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}+\left(1-{\delta }_{n}\right)\left({‖{x}_{n}-z‖}^{2}+\eta \left(\eta -2\beta \right){‖B{v}_{n}-Bz‖}^{2}\right)\\ ={‖{x}_{n}-z‖}^{2}-{\delta }_{n}\lambda \left(2\alpha -\lambda \right){‖A{u}_{n}-Az‖}^{2}\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}-\left(1-{\delta }_{n}\right)\eta \left(2\beta -\eta \right){‖B{v}_{n}-Bz‖}^{2},\end{array}$

it follows that

$\begin{array}{ll}\hfill {\delta }_{n}\lambda \left(2\alpha -\lambda \right){∥A{u}_{n}-Az∥}^{2}& \le {∥{x}_{n}-z∥}^{2}-{∥{y}_{n}-z∥}^{2}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\left(1-{\delta }_{n}\right)\eta \left(2\beta -\eta \right){∥B{v}_{n}-Bz∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le \left(∥{x}_{n}-z∥+∥{y}_{n}-z∥\right)∥{y}_{n}-{x}_{n}∥,\phantom{\rule{2em}{0ex}}\end{array}$
(3.22)

by conditions (i), (ii), λ (0, 2α) and (3.12), it implies that

$\underset{n\to \infty }{\text{lim}}∥A{u}_{n}-Az∥=0.$
(3.23)

By using the same method as (3.23), we have

$\underset{n\to \infty }{\text{lim}}∥B{v}_{n}-Bz∥=0.$
(3.24)

By nonexpansiveness of ${T}_{{r}_{n}}$, we have

$\begin{array}{c}{‖{M}_{n}-z‖}^{2}={‖{P}_{C}\left({u}_{n}-\lambda A{u}_{n}\right)-{P}_{C}\left(z-\lambda Az\right)‖}^{2}\\ \le 〈\left({u}_{n}-\lambda A{u}_{n}\right)-\left(z-\lambda Az\right),{M}_{n}-z〉\\ =\frac{1}{2}\left({‖\left({u}_{n}-\lambda A{u}_{n}\right)-\left(z-\lambda Az\right)‖}^{2}+{‖{M}_{n}-z‖}^{2}-‖\left({u}_{n}-\lambda A{u}_{n}\right)\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}{-\left(z-\lambda Az\right)-\left({M}_{n}-z\right)‖}^{2}\right)\\ \le \frac{1}{2}\left({‖{u}_{n}-z‖}^{2}+{‖{M}_{n}-z‖}^{2}-{‖\left({u}_{n}-{M}_{n}\right)-\lambda \left(A{u}_{n}-Az\right)‖}^{2}\right)\\ =\frac{1}{2}\left({‖{T}_{{r}_{n}}{x}_{n}-{T}_{{r}_{n}}z‖}^{2}+{‖{M}_{n}-z‖}^{2}-{‖{u}_{n}-{M}_{n}‖}^{2}\\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}+2\lambda 〈{u}_{n}-{M}_{n},A{u}_{n}-Az〉-{\lambda }^{2}{‖A{u}_{n}-Az‖}^{2}\right)\\ \le \frac{1}{2}\left({‖{x}_{n}-z‖}^{2}+{‖{M}_{n}-z‖}^{2}-{‖{u}_{n}-{M}_{n}‖}^{2}+2\lambda 〈{u}_{n}-{M}_{n},A{u}_{n}-Az〉\end{array}$